MGRASP and MSAL Algorithms for Centralized Traffic

MGRASP and MSAL Algorithms for Centralized Traffic
Management of Large Wireless Sensor Networks
Maria Magdalena Czajko1 , 2 and Jacek Wojciechowski1
1
2
Warsaw University of Technology, Institute of Radioelectronics, Warsaw, Poland,
e-mail: [email protected]
e-mail: [email protected]
Telekomunikacja Polska S.A., Warsaw, Poland, email: [email protected]
Abstract. This paper concerns a hop and degree constrained minimum spanning forest
problem with the minimization of the number of trees (HDMSFMT). HDMSFMT arises
during the calculation of the shortest paths in large wireless sensor networks (LWSNs)
with centralized traffic management. This is a bi-criteria optimization problem, which is
shown in this paper to be NP-hard. We formulate the HDMSFMT problem using formal
mathematical notations. Two efficient heuristic algorithms for HDMSFMT are proposed
and compared, namely Multi-objective Simulated Allocation (MSAL) and Multi-objective
Greedy Randomized Adaptive Search Procedure (MGRASP). Numerical examples show
that it is possible to achieve a set of feasible and satisfactory results for a wide range of
graphs’ types.
1 Introduction
LWSNs are invaluable in many civil applications, e.g. for monitoring of natural phenomena like
earthquakes, volcanic activity, landslides, oceanic events or biological activity of living plants.
They are also useful in military applications, e.g. battlefield monitoring. LWSNs are used for
collecting, processing, and disseminating wide ranges of complex environmental data.
A typical wireless sensor network usually consists of hundreds to thousands of spatially
distributed, autonomous, and low-power devices with limited computational capabilities using
sensors to monitor environmental or physical conditions at different, unattended locations.
Hierarchical structure has been considered as a necessity for large-scale systems. For this reason,
all sensor nodes can play a sensing and/or gateway role. The sensing node can monitor
environmental or physical conditions and collect data, whereas the gateway node can aggregate
and process the data collected from all sensing nodes and send the data to a sink node. Sensors
are usually powered by batteries with limited capacity, so the problem of energy economy is of
paramount importance. We assume that sensor nodes posses all features, i.e. memory, interfaces
or protocols we would like to use in our LWSN or it is possible to implement them easily.
In the literature we can find several hardware/software system architectures. We would like
to focus on different approaches for traffic management problem for LWSNs. There are two
basic approaches for traffic management in LWSNs: centralized [5], [6] and decentralized [4].
Decentralized scenarios assume that all sensor nodes take part in gateway selection, routing
paths’ calculation, etc. In case of centralized approaches we can make use of a high-energy
control center or base station to determine the aforementioned parameters. Experimental results
show that centralized scenarios perform better than decentralized. The key reason is that a control
center utilizes its global knowledge of the network structure to produce routing paths or
gateways’ locations that require less energy for the data transmission. The above advantages
encouraged us to focus on the centralized approach.
The next step after elaborating the communication approach is to work out an algorithm for
efficient traffic management. The researchers propose generally two scenarios [5], [6]:
clustering-based approaches (most popular), and multi-hop routing, which is partially based on
routing schemes used in wired packet networks. In the clustering-based approach sensor nodes
are grouped into clusters. Each cluster includes a dedicated head node, which processes data
gathered directly from sensor nodes within this cluster range. The head node then transmits
aggregated data to base station either directly [5] or indirectly, using routing algorithm between
cluster heads [6]. The role of a cluster head may change randomly depending on the current
energy level of sensor nodes, what leads to the improvement of power management [6].
Comparing the multi-hop and clustering based approaches we can find that the first is more
power economical because sensor nodes can communicate with each other before sending data to
the head node. In case of clusters, all sensor nodes within each cluster send data directly to
cluster head. It can lead to higher overall power consumption than in the case of multi-hop
solutions. From the reliability perspective, the clustering-based approaches perform better than
the multi-hop solutions. The reason is that, in case of clustering-based scenarios, a damage on a
sensor node has lower impact on data transmission than in case of multi-hop approaches.
To achieve the compromise between the two routing schemes compared above we propose a
tree-based routing approach with hop and degree constraints. Our solution is based on the
centralized gateway nodes’ election and determining a “forest” of sensor nodes. We assume that
each sensor node is capable of sensing and gateway functionality. Gateway nodes send data
directly to the control center. The developed architecture is presented in Fig. 1.
Figure 1. Reference architecture for LWSN.
In section 2 we describe the problem of the hop and degree constrained minimum spanning
forest with the minimization of the number of trees (HDMSFMT), which is met during
calculation of the shortest paths in our traffic management scheme. In section 3 we discuss the
complexity of HDMSFMT. Sections 4 and 5 describe two proposed heuristic algorithms: MSAL
and MGRASP followed by numerical results. Section 6 includes a test example and numerical
results while section 7 contains concluding remarks.
2 Problem Description
Consider undirected graph G=(V, E) with set V of vertices representing sensor nodes and set E of
edges representing wireless links between these nodes. All nodes in G are of the same type and
their geographical locations are known. Undirected edge e connects two nodes i and j and is
denoted by {i,j}. Cost cij representing a distance between nodes i and j is assigned to each edge.
Tree Tn=(VTn, ETn), where n=1, 2, ..., N, is a connected acyclic subgraph of G, where VTn ⊆ V
and ETn ⊆ E. ETn contains edges forming a tree spanned over vertices from VTn. In VTn we
distinguish one node r, which represents a root (gateway node) of Tn. Spanning forest F = {T1, T2,
..., TN} is a set of N mutually vertex disjoint trees Tn (n = 1, 2, ..., N), i.e. U Nn=1 VTn = V, n≠m ⇒
VTn ∩ VTm = ∅.
Let’s replace each edge e in E by two parallel arcs in opposite directions. We obtain then a
bidirected graph GB=(V,A), where A represents the set of arcs, i.e. A={(i,j): {i,j}∈E}. We assume
that directed forest FB of graph GB consists of directed trees TBn. Tree TBn is obtained from Tn by
assigning orientation to each edge towards root node. In such a transformed model, root nodes
have only incoming edges while other nodes have one outgoing arc (see Fig. 2). Assume also that
cij denotes the cost of arcs (i,j) and (j,i) linking vertices i and j. It is the same constant as in case
of the undirected graph model.
Figure 2. Graph and forest.
The optimization problem that we define below is constrained. H is the maximum acceptable
distance measured as the number of hops between the root and vertices in TBn (n = 1, 2, ..., N). In
our model k is the level of each arc in TBn (see Fig. 2). D is the maximum acceptable value of the
sum of outdegree and indegree of vertex in FB.
The objective of the HDMSFMT problem is to find a spanning forest FB in GB with the
minimum number N of trees and the minimum total cost under D and H constraints. We state the
mathematical formulation of HDMSFMT below:
indices
i, j, l, m, n = 1, 2, ..., N
k = 1, 2, ..., H
constants
cij
cost of edge (i,j)
nodes
levels
D
H
maximum degree (sum of indegree and outdegree) of each node
maximum length of a path between root and a non-root node in tree TBn
variables
yn
= 1 if node n is a root of tree TBn, 0 otherwise
= 1 if directed edge (i,j) of graph GB is on level k in tree TBn, 0 otherwise
xijk
objectives
f1:
minimize
∑ i ∑ j c ij ∑ k x ijk
(1)
f2:
∑n y n
(2)
minimize
constraints
∑ i x ij1 − D ⋅ y n ≤ 0
n = j = 1, 2, ..., N
(3)
∑ j ∑ k x ijk + y n = 1
n = i = 1, 2, ..., N
(4)
x ijk + ∑ l x lm(k +1) ≤ D
i, j = 1, 2, ..., N
i=m
k = 1, 2, ..., H-1
x ij(k +1) − ∑ m x lmk ≤ 0
i, j = 1, 2, ..., N
l=j
k = 1, 2, ..., H-1
∑ i ∑ j ∑ k x ijk + ∑ n y n = N
(5)
(6)
(7)
Objective function f1 (1) returns the total cost of FB, which is the sum of costs assigned to the
edges that form FB. The second objective function f2 (2) returns the total number of trees in FB.
Inequality (3) assures that if node n is not a root, then there is no arc directed to n, which is on the
first level (see Fig. 2). Moreover, when n is a root node, then maximum degree of n can not
exceed D. Constraint (4) forces only one outgoing edge at n when n is not a root node and the
lack of outgoing edges if n is a root node. Inequality (5) assures that the sum of incoming edges
at node i, which are on level k+1, and outgoing edges at node i, which are on level k, should not
exceed D. Existence of an incoming edge at node j on level k+1 implies the existence of outgoing
edge at node j on level k (6). Equation (7) defines the relationship between the number of trees
and the number of nodes and edges in the forest.
3 Complexity of HDMSFMT Problem
The hop constrained [3] with the degree-constrained minimum spanning tree problem [2] is NPhard and can be reduced to HDMSFMT, so the HDMSFMT problem is also NP-hard.
The HDMSFMT problem is additionally complicated by the fact that it belongs to a set of
multi-criteria optimization problems. Multi-objective optimization is the process of searching one
or more decision variables that simultaneously satisfy all constraints, and optimize an objective
function vector [1]. In our case we have two objectives: the minimization of the total cost and the
minimization of the number of trees in the forest, which conflict with each other.
While solving a multi-criteria problem we obtain not only one solution vector of decision
variables. It will be a set of decision vectors, which is known as Pareto-optimal set. Paretooptimal solutions are not dominated by any other solution in the feasible space. A solution x is
dominated if there exists a feasible solution y that is at least as good as x with respect to every
dimension of objective space and strictly better than x on at least one objective. Thus, a nondominated solution is any solution that is not dominated by any other feasible solution.
The Pareto-optimal frontier represents images corresponding to Pareto-optimal solutions in
the objective space and is formed by the solutions whose performance on one objective cannot be
improved without sacrificing performance on at least one other [2], [7], and [8].
As our problem is NP-hard we propose two heuristics presented below to solve it.
4 MSAL Algorithm
A Multi-objective Simulated Allocation algorithm (MSAL) was developed to solve HDMSFMT.
A single-objective version of Simulated Allocation (SAL) is presented in [7].
Generally, MSAL uses two procedures: allocate and disconnect. The procedure allocate
randomly selects nodes to classify them as root or non-root nodes in forest Fi taking into
account edges’ existence, cost, and D and H constraints. The procedure disconnect removes a
defined number of nodes from the forest in a random way (percentage parameter). Both
operations are run sequentially. The pseudo-code of MSAL is presented below:
0: initialize: G=(V,E), Pareto=∅, step, limit, H, D, percentage, i=1
1: allocate all nodes to create a forest Fi
2: add this solution to Pareto
3: repeat
4: i++
5: disconnect randomly a percentage of all nodes from Fi
6: allocate all nodes to create forest Fi
7: if new solution is non-dominated then update Pareto
8: step++
9: until step = limit
5 MGRASP Algorithm
A Multi-objective Greedy Adaptive Search Procedure (MGRASP) was developed as an
alternative method to solve HDMSFMT. MGRASP is divided into two phases that are run
sequentially, predefined number of times. The first phase (lines: 3-11) allows obtaining as low
number of trees as possible and a low value of the total cost of the forest, not necessary optimal,
and satisfying the constraints. The second phase (lines: 12-16) improves the results.
0: initialize: Pareto=∅, H, D, step1=0, limit1, step2=0, limit2, p
1: repeat
2: initialize: B=∅, G=(V,E), Ti=∅, i=0
3: while V ≠ ∅ do
4:
i++
5:
choose the root node of tree Ti, e.g. randomly
6:
determine spanning tree Ti of G while satisfying constraints D and H
7:
add the remaining nodes and edges that do not create Ti to backup set B
8:
fill Ti up with nodes from B with probability p taking into account H, D
9:
overwrite graph G=(V,E) by backup set B
10: endwhile
11: if a new solution is non-dominated then update Pareto
12: repeat
13:
choose randomly two nodes to be exchanged or to switch the branch
14:
if a new solution is non-dominated then update Pareto
15:
step2++
16: until step2 = limit2
17: step1++
18: until step1 = limit1
6 Computational Results
To evaluate quality of both algorithms numerical experiments were carried out. The
experiments were performed on the test graphs of the following types:
- graphs with 50 nodes and 100 edges (connected), and 100 nodes and 200 edges
(disconnected); the edges were generated randomly,
- connected graphs with 50 nodes and 613 edges, and 100 nodes and 2475 edges; the edges
were generated randomly.
Costs of links (weights of edges) were generated randomly from the (0, 100> interval. Also
the root nodes were selected randomly. Simulations were performed for three combinations of
the values of H and D constraints: D=2 and H=4, D=4 and H=3, D=6 and H=5. In the first
phase of MGRASP the probability p of the node linking to a tree, was set to 0, 0.3, 0.6 and 1.
Limit1 and limit2 parameters were set to 100 and 50000, respectively. The values of
percentage parameter for MSAL that we checked were set to 10%, 50%, and 80%. The limit
parameter was 50000.
Table 1 includes the number of pseudo-Pareto-optimal solutions found by both of the
proposed algorithms. Light grey shaded records in the table show the highest values of the
parameters for each case (graph type, D and H combination) among all values obtained by
MSAL and MGRASP, respectively. It can be observed that MSAL is more effective than
MGRASP because it provides larger size of the pseudo-Pareto set. As expected, MSAL with
percentage=50% and 80% allowed obtaining a larger size of pseudo-Pareto set than MSAL
with percentage=10% because of higher diversification.
Table 1. Number of Pseudo-Pareto-optimal solutions and domination parameter.
No. of Pseudo-Pareto-Optimal Solutions
type of G
MSAL
D, H
10% 50% 80% p = 0
50n,100e
50n,613e
100n,200e
100n,2475e
Domination Parameter
MGRASP
0,3
0,6
MSAL
1
MGRASP
10% 50% 80% p = 0 0,3
0,6
1
2, 4
12
10
12
8
6
6
7
9
3
1
0
1
3
3
4, 3
11
13
14
6
7
7
8
7
3
6
6
0
1
0
6, 5
14
17
16
5
3
3
3
9
2
6
5
0
0
0
2, 4
2
3
2
2
3
3
3
2
1
0
1
1
0
0
4, 3
3
4
3
4
2
2
2
3
0
0
4
1
0
0
6, 5
3
3
2
2
1
1
1
2
1
0
2
0
0
0
2, 4
14
12
12
7
8
7
5
14
1
0
3
3
2
1
4, 3
13
13
17
8
7
3
7
7
6
5
7
0
1
0
6, 5
15
16
16
5
3
4
4
12
5
4
5
1
0
0
2, 4
2
1
2
1
2
1
3
1
1
0
1
0
0
0
4, 3
3
3
2
3
3
2
2
2
0
1
3
0
2
0
6, 5
2
3
3
2
1
1
1
2
1
0
0
0
1
0
It can happen that one algorithm provides many pseudo-Pareto-optimal solutions but most
of them are dominated by solutions obtained thanks to the other algorithm. Thus, we proposed
and checked an additional parameter, a domination parameter, which allows for comparing the
algorithms more precisely. The domination parameter defines the number of non-dominated
solutions generated by the algorithm out of pseudo-Pareto-optimal solutions obtained by all
compared methods. It could be seen from Table 1 that MSAL with percentage=10% and
MGRASP with p=0 produces a higher value of the dominance parameter than in the remaining
cases. It means, that MSAL with percentage=10% provides the highest number of nondominated solutions among all solutions with percentage=10%, 50%, and 80%. Similar
situation is observed in case of MGRASP. It seems that too strong diversification in MSAL by
disconnecting a high number of nodes (disconnect procedure) leads to worse solutions. In case
of MGRASP we observe, that filling in the trees with as many nodes as possible gives worse
solutions than in case of MGRASP with p=0.
In the next step we calculate the domination parameter to compare MSAL with
percentage=10% and MGRASP with p=0, as the best values of domination parameter were
obtained in these cases. It can be observed in Table 2 that the domination parameter is the
highest for MSAL with percentage=10% for sparse graphs (50 nodes, 100 edges; 100 nodes,
200 edges), what means that MSAL provides the best solutions. On the other hand MGRASP
allowed obtaining better solutions for denser graphs (50 nodes, 613 edges; 100 nodes, 2475
edges). But generally, MSAL provides the highest values of the domination parameter.
Table 2. Domination for MSAL
(percentage=10%) and MGRASP (p=0)
2900
type of G
50n,100e
50n,613e
100n,200e
100n,2475e
D, H MSAL MGRASP
10%
p=0
2, 4
12
0
4, 3
10
1
6, 5
12
3
2, 4
0
2
4, 3
3
0
6, 5
0
2
2, 4
11
4
4, 3
13
2
6, 5
12
4
2, 4
0
1
4, 3
2
3
6, 5
1
1
the total cost of the forest
Domination
2700
2500
2300
2100
1900
1700
1500
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
the number of trees
MSAL - 10%
MGRASP - p=0
Figure 3. Example of the pseudo-Pareto frontier for
MSAL with percentage = 10% and MGRASP with p = 0.
Graph with 100 nodes, 200 edges, D = 2 and H = 4. The
number of trees in the forest is assigned to the horizontal
axis and the total cost of the forest is assigned to the
vertical axis.
Fig. 3 shows an example of the pseudo-Pareto frontier for MSAL with percentage=10%,
and MGRASP with p=0. It can be seen that MSAL gives more pseudo-Pareto-optimal
solutions than MGRASP. On the other hand MGRASP sometimes allows obtaining lower cost
for the minimum number of trees than MSAL as we can see in Fig. 3.
7 Conclusions
In this paper we defined the so-called hop and degree constrained minimum spanning forest
problem with the minimization of the number of trees (HDMSFMT), which could be met during
the shortest paths’ calculation in centralized managed LWSNs. We have not found any
formulation of this problem in the literature known to us. Since HDMSFMT is NP-hard, we
developed two efficient multi-objective heuristics: MSAL and MGRASP algorithms. Numerical
examples show that it is possible to achieve a set of feasible and satisfactory results for a wide
range of graphs’ types. Generally, the best solutions were obtained by the MSAL algorithm with
percentage=10%, but it sometimes can be observed that MGRASP allows obtaining lower cost
for the minimum number of trees than MSAL.
Further research will be to improve the methods presented in this paper i.e. a tabu-list can be
implemented to forbid selection of solutions got in previous iterations or additional new rules for
root selection. Adding a condition to build as balanced spanning forest as possible could also
extend the analyzed problem. Another task will be to develop decision algorithm to choose the
most preferred solution from the pseudo-Pareto-optimal set.
Bibliography
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
Baños, R., Gil, C., Paechter, B., Ortega, J. Parallelization of population-based multiobjective meta-heuristics: An empirical study. Applied Mathematical Modelling 30
(2006) 578-592.
Da Cunha, A. S., Lucena, A. Algorithms for the degree-constrained minimum spanning
tree problem. Electronic Notes in Discrete Mathematics 19 (2005) 403-409.
Gouveia, L. Multicommodity flow models for spanning trees with hop constraints.
European Journal of Operational Research 95 (1996) 178-190.
Heinzelman, W. B., Chandrakasan, A. P., Balakrishnan, H. Energy-efficient
communication protocol for wireless microsensor networks. Proc. 33rd Hawaii Int’l.
Conf. Sys. Sci. (2000).
Heinzelman, W. B., Chandrakasan, A. P., Balakrishnan, H. An application-specific
protocol architecture for wireless microsensor networks. IEEE Transactions on Wireless
Communications (2002) vol. 1, Issue 4, 660 – 670.
Muruganathan, S. D., Ma, D. C., Bhasin, R. I., Fapojuwo, A. O. A centralized energyefficient routing protocol for wireless sensor networks. IEEE Communications Magazine
(2005), vol. 43, Issue 3, S8 – 13.
Pióro, M., Medhi, D. Routing, flow, and capacity design in communication and computer
networks. Elsevier, 2004.
Rahimi-Vahed, A. R., Rabbani, M., Tavakkoli-Moghaddam, R., Torabi, S. A., Jolai, F. A
multi-objective scatter search for a mixed-model assembly line sequencing problem.
Advanced Engineering Informatics 21 (2007) 85-99.